From Nookipedia, the Animal Crossing wiki
The Auction House is a building that appears in the City in Animal Crossing: City Folk, run by a gyroid called Lloid. It is the only building in the City open for business all day and every day. The Auction House mainly acts as a place to sell furniture or other items to friends visiting over Wi-Fi or human residents in town. Auctions happen over a set period (displaying and bidding is only allowed during weekends; each weekend alternates between the two) which Lloid will mention when talking to him. The Auction House has nine dark brown display tables where items will be shown. Three of these tables are bigger than the other six.
An item sold at the Auction House will be sent to the buyer's mailbox, and the seller will have the money sent to their savings account. However, if an item is not sold, the item will be mailed back to the player by Lloid.
The Auction House can also be used to store furniture for later use, and Lloid can also let the player access his or her closet through the "checkroom" to obtain items to sell at an auction.
The termination of the Nintendo Wi-Fi Connection means that the Auction House now has little use, except for trading items between human residents living in town.
The Auction House was effectively replaced by Re-Tail in Animal Crossing: New Leaf.